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Mirrors > Home > NFE Home > Th. List > imakeq2 | GIF version |
Description: Equality theorem for Kuratowski image. (Contributed by SF, 12-Jan-2015.) |
Ref | Expression |
---|---|
imakeq2 | ⊢ (A = B → (C “k A) = (C “k B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexeq 2808 | . . 3 ⊢ (A = B → (∃y ∈ A ⟪y, x⟫ ∈ C ↔ ∃y ∈ B ⟪y, x⟫ ∈ C)) | |
2 | 1 | abbidv 2467 | . 2 ⊢ (A = B → {x ∣ ∃y ∈ A ⟪y, x⟫ ∈ C} = {x ∣ ∃y ∈ B ⟪y, x⟫ ∈ C}) |
3 | df-imak 4189 | . 2 ⊢ (C “k A) = {x ∣ ∃y ∈ A ⟪y, x⟫ ∈ C} | |
4 | df-imak 4189 | . 2 ⊢ (C “k B) = {x ∣ ∃y ∈ B ⟪y, x⟫ ∈ C} | |
5 | 2, 3, 4 | 3eqtr4g 2410 | 1 ⊢ (A = B → (C “k A) = (C “k B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∈ wcel 1710 {cab 2339 ∃wrex 2615 ⟪copk 4057 “k cimak 4179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rex 2620 df-imak 4189 |
This theorem is referenced by: imakeq2i 4227 imakeq2d 4229 addceq1 4383 phieq 4570 opeq1 4578 opeq2 4579 proj1eq 4589 proj2eq 4590 |
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